Calculating the area of a curve rotating around an axis

Calculate the area of the rotational surface that arise when the curve rotates around the y-axis.

My approach is something along the lines of:
Use an integral to calculate the area.
Since the curve is rotating around the y-axis, we need to (not sure about the english term), "extract" x from the formula (we need to get the height between the curve and the y-axis).

Since the curve is symmetrical around the y-axis we don't have to care the negative x; we simply multiply by 2 to get the total area.

You found the area of the region, not the surface area of the surface of revolution. Have you learned the formula for the surface area of a surface of revolution? If not, then here is an article about surfaces of revolution.